IMO Shortlist 2010 problem G4

Given a triangle , with as its incenter and as its circumcircle, intersects again at . Let be a point on the arc , and a point on the segment , such that . If is the midpoint of , prove that the meeting point of the lines and lies on .

Proposed by Tai Wai Ming and Wang Chongli, Hong Kong

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Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$.
Proposed by Tai Wai Ming and Wang Chongli, Hong Kong